Assignment 1 - Exercise 4 b

Exercise 4.b - testing by simulation

# 4) ----------------------------------------------------------------------
n <- 1e4
#Covariance matrix from Exercise 3
sigma <- matrix(c(1,-1, -1,
                  -1, 2, 3,
                  -1, 3, 6), nrow = 3, byrow = T)
sigma

##      [,1] [,2] [,3]
## [1,]    1   -1   -1
## [2,]   -1    2    3
## [3,]   -1    3    6

#simulating (X,Y,Z)
xyz <- rmvnorm(n = n, sigma = sigma)
y <-xyz[,2]

#our best beta from assignment
b <- c(-3, 2)/5

#Expected squared error - function
mse <- function(b){
  mean((y - b%*%t(xyz[ ,c(1,3)]) )^2 )
}
#test 
(min <- mse(b = b))

## [1] 0.2010512

#grid of beta's
b1 <- seq(from = -3, to = 3, length.out = 1e2)
b2 <- seq(from = -5, to = 5,length.out = 1e2)
b_grid <- expand.grid(b1 = b1, b2 = b2)

#add mse for each beta vector
mse_vec <- apply(X = b_grid, MARGIN = 1, FUN = function(vec) mse(b = vec))
#set as matrix for plotting
mse_mat <- matrix(data = mse_vec, nrow = length(b1), ncol = length(b2), byrow = T)
#plot
plot_ly(x = b1, y = b2, z = mse_mat) %>% add_surface() %>% add_markers(x = b[1], y = b[2], z = min)